Recall that the set of fixed tensor train rank tensors forms a smooth embedded submanifold \({\cal M}_{\mathbf {r}}\)  of dimension \(\sum _{k= 1}^d r_{k-1}n_kr_k-\sum _{k = 1}^{d-1}r_k^2\) .
Let \(T_{{\cal X}}{\cal M}_{\mathbf {r}}\)  be the tangent space of \({\cal M}_{\mathbf {r}}\)  at \({\cal X}\) .
For an objective function \(h\left({\cal X}\right)= \frac{1}{2}\left\Vert  F\left({\cal X}\right)\right\Vert _{{\footnotesize {\mathsf {F}}}}^2\)  defined on \({\cal M}_{\mathbf {r}}\) , the Gauss-Newton direction \(\xi \in T_{{\cal X}}{\cal M}_{\mathbf {r}}\)  is the solution of the following Gauss-Newton equation [1]}
\(\left\langle \mathrm {D} F\left({\cal X}\right)\left[\eta \right],F\left({\cal X}\right)\right\rangle + \left\langle \mathrm {D} F\left({\cal X}\right)\left[\eta \right],\mathrm {D} F\left({\cal X}\right)\left[\xi \right]\right\rangle = 0,~\text{for all}~\eta \in T_{{\cal X}}{\cal M}_{\mathbf {r}}.\) 
